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Theorem ressid 13444
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressid  |-  ( W  e.  X  ->  ( Ws  B )  =  W )

Proof of Theorem ressid
StepHypRef Expression
1 ssid 3303 . 2  |-  B  C_  B
2 ressid.1 . . 3  |-  B  =  ( Base `  W
)
3 fvex 5675 . . 3  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2450 . 2  |-  B  e. 
_V
5 eqid 2380 . . 3  |-  ( Ws  B )  =  ( Ws  B )
65, 2ressid2 13437 . 2  |-  ( ( B  C_  B  /\  W  e.  X  /\  B  e.  _V )  ->  ( Ws  B )  =  W )
71, 4, 6mp3an13 1270 1  |-  ( W  e.  X  ->  ( Ws  B )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2892    C_ wss 3256   ` cfv 5387  (class class class)co 6013   Basecbs 13389   ↾s cress 13390
This theorem is referenced by:  submid  14671  subgid  14866  gaid2  15000  subrgid  15790  rlmsca  16191  rlmsca2  16192  pjff  16855  rlmbn  19175  ishl2  19184  evlrhm  19806  evl1sca  19810  evl1var  19812  pf1ind  19835  dchrptlem2  20909  lnmfg  26842  lmhmfgsplit  26846  pwslnmlem2  26857  dsmmfi  26866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-ress 13396
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